LMFDB - Embedded newform 2704.2.f.o.337.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2704,2,Mod(337,2704)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2704.337"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2704, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2704 = 2^{4} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2704.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,4,0,0,0,0,0,-6,0,0,0,0,0,0,0,4,0,0,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$21.5915487066$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 169)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.3
Root			$-1.80194i$ of defining polynomial
Character	$\chi$	$=$	2704.337
Dual form			2704.2.f.o.337.4

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.554958 q^{3} -1.44504i q^{5} +2.04892i q^{7} -2.69202 q^{9} -2.55496i q^{11} -0.801938i q^{15} +5.29590 q^{17} +5.85086i q^{19} +1.13706i q^{21} -1.89008 q^{23} +2.91185 q^{25} -3.15883 q^{27} +2.26875 q^{29} +4.26875i q^{31} -1.41789i q^{33} +2.96077 q^{35} -5.35690i q^{37} +1.27413i q^{41} +6.13706 q^{43} +3.89008i q^{45} -2.95108i q^{47} +2.80194 q^{49} +2.93900 q^{51} +5.52111 q^{53} -3.69202 q^{55} +3.24698i q^{57} -12.2078i q^{59} +8.56465 q^{61} -5.51573i q^{63} -0.576728i q^{67} -1.04892 q^{69} +4.59419i q^{71} +10.5526i q^{73} +1.61596 q^{75} +5.23490 q^{77} +15.7778 q^{79} +6.32304 q^{81} -7.72348i q^{83} -7.65279i q^{85} +1.25906 q^{87} -6.61356i q^{89} +2.36898i q^{93} +8.45473 q^{95} +11.9269i q^{97} +6.87800i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 6 q^{9} + 4 q^{17} - 10 q^{23} + 10 q^{25} - 2 q^{27} - 2 q^{29} - 8 q^{35} + 26 q^{43} + 8 q^{49} - 2 q^{51} + 2 q^{53} - 12 q^{55} + 8 q^{61} + 12 q^{69} + 30 q^{75} - 16 q^{77} + 10 q^{79}+ \cdots + 6 q^{95}+O(q^{100}) $ 6 * q + 4 * q^3 - 6 * q^9 + 4 * q^17 - 10 * q^23 + 10 * q^25 - 2 * q^27 - 2 * q^29 - 8 * q^35 + 26 * q^43 + 8 * q^49 - 2 * q^51 + 2 * q^53 - 12 * q^55 + 8 * q^61 + 12 * q^69 + 30 * q^75 - 16 * q^77 + 10 * q^79 - 2 * q^81 + 36 * q^87 + 6 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times$.

$n$	$677$	$1185$	$2367$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.554958	0.320405	0.160203	−	0.987084i	$-0.448785\pi$
$3$	0.554958	0.320405	0.160203	+	0.987084i	$0.448785\pi$
$4$	0	0
$5$	− 1.44504i	− 0.646242i	−0.946358	−	0.323121i	$-0.895268\pi$
$5$	− 1.44504i	− 0.646242i	0.946358	−	0.323121i	$-0.104732\pi$
$6$	0	0
$7$	2.04892i	0.774418i	0.921992	+	0.387209i	$0.126561\pi$
$7$	2.04892i	0.774418i	−0.921992	+	0.387209i	$0.873439\pi$
$8$	0	0
$9$	−2.69202	−0.897340
$10$	0	0
$11$	− 2.55496i	− 0.770349i	−0.922844	−	0.385174i	$-0.874141\pi$
$11$	− 2.55496i	− 0.770349i	0.922844	−	0.385174i	$-0.125859\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 0.801938i	− 0.207059i
$16$	0	0
$17$	5.29590	1.28444	0.642222	−	0.766519i	$-0.278013\pi$
$17$	5.29590	1.28444	0.642222	+	0.766519i	$0.278013\pi$
$18$	0	0
$19$	5.85086i	1.34228i	0.741331	+	0.671139i	$0.234195\pi$
$19$	5.85086i	1.34228i	−0.741331	+	0.671139i	$0.765805\pi$
$20$	0	0
$21$	1.13706i	0.248128i
$22$	0	0
$23$	−1.89008	−0.394110	−0.197055	−	0.980392i	$-0.563138\pi$
$23$	−1.89008	−0.394110	−0.197055	+	0.980392i	$0.563138\pi$
$24$	0	0
$25$	2.91185	0.582371
$26$	0	0
$27$	−3.15883	−0.607918
$28$	0	0
$29$	2.26875	0.421296	0.210648	−	0.977562i	$-0.432443\pi$
$29$	2.26875	0.421296	0.210648	+	0.977562i	$0.432443\pi$
$30$	0	0
$31$	4.26875i	0.766690i	0.923605	+	0.383345i	$0.125228\pi$
$31$	4.26875i	0.766690i	−0.923605	+	0.383345i	$0.874772\pi$
$32$	0	0
$33$	− 1.41789i	− 0.246824i
$34$	0	0
$35$	2.96077	0.500462
$36$	0	0
$37$	− 5.35690i	− 0.880668i	−0.897834	−	0.440334i	$-0.854860\pi$
$37$	− 5.35690i	− 0.880668i	0.897834	−	0.440334i	$-0.145140\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.27413i	0.198985i	0.995038	+	0.0994926i	$0.0317220\pi$
$41$	1.27413i	0.198985i	−0.995038	+	0.0994926i	$0.968278\pi$
$42$	0	0
$43$	6.13706	0.935893	0.467947	−	0.883757i	$-0.344994\pi$
$43$	6.13706	0.935893	0.467947	+	0.883757i	$0.344994\pi$
$44$	0	0
$45$	3.89008i	0.579899i
$46$	0	0
$47$	− 2.95108i	− 0.430460i	−0.976563	−	0.215230i	$-0.930950\pi$
$47$	− 2.95108i	− 0.430460i	0.976563	−	0.215230i	$-0.0690501\pi$
$48$	0	0
$49$	2.80194	0.400277
$50$	0	0
$51$	2.93900	0.411542
$52$	0	0
$53$	5.52111	0.758382	0.379191	−	0.925318i	$-0.376202\pi$
$53$	5.52111	0.758382	0.379191	+	0.925318i	$0.376202\pi$
$54$	0	0
$55$	−3.69202	−0.497832
$56$	0	0
$57$	3.24698i	0.430073i
$58$	0	0
$59$	− 12.2078i	− 1.58931i	−0.607059	−	0.794657i	$-0.707651\pi$
$59$	− 12.2078i	− 1.58931i	0.607059	−	0.794657i	$-0.292349\pi$
$60$	0	0
$61$	8.56465	1.09659	0.548295	−	0.836285i	$-0.315277\pi$
$61$	8.56465	1.09659	0.548295	+	0.836285i	$0.315277\pi$
$62$	0	0
$63$	− 5.51573i	− 0.694917i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 0.576728i	− 0.0704586i	−0.999379	−	0.0352293i	$-0.988784\pi$
$67$	− 0.576728i	− 0.0704586i	0.999379	−	0.0352293i	$-0.0112162\pi$
$68$	0	0
$69$	−1.04892	−0.126275
$70$	0	0
$71$	4.59419i	0.545230i	0.962123	+	0.272615i	$0.0878885\pi$
$71$	4.59419i	0.545230i	−0.962123	+	0.272615i	$0.912112\pi$
$72$	0	0
$73$	10.5526i	1.23508i	0.786538	+	0.617542i	$0.211872\pi$
$73$	10.5526i	1.23508i	−0.786538	+	0.617542i	$0.788128\pi$
$74$	0	0
$75$	1.61596	0.186595
$76$	0	0
$77$	5.23490	0.596572
$78$	0	0
$79$	15.7778	1.77514	0.887569	−	0.460674i	$-0.152392\pi$
$79$	15.7778	1.77514	0.887569	+	0.460674i	$0.152392\pi$
$80$	0	0
$81$	6.32304	0.702560
$82$	0	0
$83$	− 7.72348i	− 0.847762i	−0.905718	−	0.423881i	$-0.860667\pi$
$83$	− 7.72348i	− 0.847762i	0.905718	−	0.423881i	$-0.139333\pi$
$84$	0	0
$85$	− 7.65279i	− 0.830062i
$86$	0	0
$87$	1.25906	0.134986
$88$	0	0
$89$	− 6.61356i	− 0.701036i	−0.936556	−	0.350518i	$-0.886005\pi$
$89$	− 6.61356i	− 0.701036i	0.936556	−	0.350518i	$-0.113995\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.36898i	0.245652i
$94$	0	0
$95$	8.45473	0.867437
$96$	0	0
$97$	11.9269i	1.21100i	0.795847	+	0.605498i	$0.207026\pi$
$97$	11.9269i	1.21100i	−0.795847	+	0.605498i	$0.792974\pi$
$98$	0	0
$99$	6.87800i	0.691265i
$100$	0	0
$101$	−13.0640	−1.29991	−0.649957	−	0.759971i	$-0.725213\pi$
$101$	−13.0640	−1.29991	−0.649957	+	0.759971i	$0.725213\pi$
$102$	0	0
$103$	9.16852	0.903401	0.451701	−	0.892170i	$-0.350818\pi$
$103$	9.16852	0.903401	0.451701	+	0.892170i	$0.350818\pi$
$104$	0	0
$105$	1.64310	0.160351
$106$	0	0
$107$	6.89977	0.667026	0.333513	−	0.942745i	$-0.391766\pi$
$107$	6.89977	0.667026	0.333513	+	0.942745i	$0.391766\pi$
$108$	0	0
$109$	0.121998i	0.0116853i	0.999983	+	0.00584264i	$0.00185978\pi$
$109$	0.121998i	0.0116853i	−0.999983	+	0.00584264i	$0.998140\pi$
$110$	0	0
$111$	− 2.97285i	− 0.282171i
$112$	0	0
$113$	7.30798	0.687477	0.343738	−	0.939065i	$-0.388307\pi$
$113$	7.30798	0.687477	0.343738	+	0.939065i	$0.388307\pi$
$114$	0	0
$115$	2.73125i	0.254690i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	10.8509i	0.994696i
$120$	0	0
$121$	4.47219	0.406563
$122$	0	0
$123$	0.707087i	0.0637559i
$124$	0	0
$125$	− 11.4330i	− 1.02260i
$126$	0	0
$127$	−18.9705	−1.68336	−0.841678	−	0.539980i	$-0.818432\pi$
$127$	−18.9705	−1.68336	−0.841678	+	0.539980i	$0.818432\pi$
$128$	0	0
$129$	3.40581	0.299865
$130$	0	0
$131$	−3.25667	−0.284536	−0.142268	−	0.989828i	$-0.545440\pi$
$131$	−3.25667	−0.284536	−0.142268	+	0.989828i	$0.545440\pi$
$132$	0	0
$133$	−11.9879	−1.03948
$134$	0	0
$135$	4.56465i	0.392862i
$136$	0	0
$137$	− 0.792249i	− 0.0676864i	−0.999427	−	0.0338432i	$-0.989225\pi$
$137$	− 0.792249i	− 0.0676864i	0.999427	−	0.0338432i	$-0.0107747\pi$
$138$	0	0
$139$	11.3394	0.961799	0.480899	−	0.876776i	$-0.340310\pi$
$139$	11.3394	0.961799	0.480899	+	0.876776i	$0.340310\pi$
$140$	0	0
$141$	− 1.63773i	− 0.137922i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 3.27844i	− 0.272260i
$146$	0	0
$147$	1.55496	0.128251
$148$	0	0
$149$	8.40581i	0.688631i	0.938854	+	0.344316i	$0.111889\pi$
$149$	8.40581i	0.688631i	−0.938854	+	0.344316i	$0.888111\pi$
$150$	0	0
$151$	14.1293i	1.14983i	0.818215	+	0.574913i	$0.194964\pi$
$151$	14.1293i	1.14983i	−0.818215	+	0.574913i	$0.805036\pi$
$152$	0	0
$153$	−14.2567	−1.15258
$154$	0	0
$155$	6.16852	0.495468
$156$	0	0
$157$	−9.43296	−0.752832	−0.376416	−	0.926451i	$-0.622844\pi$
$157$	−9.43296	−0.752832	−0.376416	+	0.926451i	$0.622844\pi$
$158$	0	0
$159$	3.06398	0.242990
$160$	0	0
$161$	− 3.87263i	− 0.305206i
$162$	0	0
$163$	8.70410i	0.681758i	0.940107	+	0.340879i	$0.110725\pi$
$163$	8.70410i	0.681758i	−0.940107	+	0.340879i	$0.889275\pi$
$164$	0	0
$165$	−2.04892	−0.159508
$166$	0	0
$167$	− 23.8538i	− 1.84587i	−0.384961	−	0.922933i	$-0.625785\pi$
$167$	− 23.8538i	− 1.84587i	0.384961	−	0.922933i	$-0.374215\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 15.7506i	− 1.20448i
$172$	0	0
$173$	18.8552	1.43353	0.716766	−	0.697314i	$-0.245622\pi$
$173$	18.8552	1.43353	0.716766	+	0.697314i	$0.245622\pi$
$174$	0	0
$175$	5.96615i	0.450998i
$176$	0	0
$177$	− 6.77479i	− 0.509224i
$178$	0	0
$179$	6.02177	0.450088	0.225044	−	0.974349i	$-0.427747\pi$
$179$	6.02177	0.450088	0.225044	+	0.974349i	$0.427747\pi$
$180$	0	0
$181$	4.77777	0.355129	0.177565	−	0.984109i	$-0.443178\pi$
$181$	4.77777	0.355129	0.177565	+	0.984109i	$0.443178\pi$
$182$	0	0
$183$	4.75302	0.351353
$184$	0	0
$185$	−7.74094	−0.569125
$186$	0	0
$187$	− 13.5308i	− 0.989470i
$188$	0	0
$189$	− 6.47219i	− 0.470782i
$190$	0	0
$191$	−18.4306	−1.33359	−0.666795	−	0.745242i	$-0.732334\pi$
$191$	−18.4306	−1.33359	−0.666795	+	0.745242i	$0.732334\pi$
$192$	0	0
$193$	− 6.05429i	− 0.435798i	−0.975971	−	0.217899i	$-0.930080\pi$
$193$	− 6.05429i	− 0.435798i	0.975971	−	0.217899i	$-0.0699203\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 11.4155i	− 0.813321i	−0.913579	−	0.406660i	$-0.866693\pi$
$197$	− 11.4155i	− 0.813321i	0.913579	−	0.406660i	$-0.133307\pi$
$198$	0	0
$199$	−13.9051	−0.985710	−0.492855	−	0.870111i	$-0.664047\pi$
$199$	−13.9051	−0.985710	−0.492855	+	0.870111i	$0.664047\pi$
$200$	0	0
$201$	− 0.320060i	− 0.0225753i
$202$	0	0
$203$	4.64848i	0.326259i
$204$	0	0
$205$	1.84117	0.128593
$206$	0	0
$207$	5.08815	0.353651
$208$	0	0
$209$	14.9487	1.03402
$210$	0	0
$211$	13.2446	0.911795	0.455897	−	0.890032i	$-0.349318\pi$
$211$	13.2446	0.911795	0.455897	+	0.890032i	$0.349318\pi$
$212$	0	0
$213$	2.54958i	0.174694i
$214$	0	0
$215$	− 8.86831i	− 0.604814i
$216$	0	0
$217$	−8.74632	−0.593739
$218$	0	0
$219$	5.85623i	0.395727i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	7.33513i	0.491196i	0.969372	+	0.245598i	$0.0789844\pi$
$223$	7.33513i	0.491196i	−0.969372	+	0.245598i	$0.921016\pi$
$224$	0	0
$225$	−7.83877	−0.522585
$226$	0	0
$227$	8.67456i	0.575751i	0.957668	+	0.287875i	$0.0929489\pi$
$227$	8.67456i	0.575751i	−0.957668	+	0.287875i	$0.907051\pi$
$228$	0	0
$229$	13.6866i	0.904439i	0.891907	+	0.452219i	$0.149368\pi$
$229$	13.6866i	0.904439i	−0.891907	+	0.452219i	$0.850632\pi$
$230$	0	0
$231$	2.90515	0.191145
$232$	0	0
$233$	5.08815	0.333336	0.166668	−	0.986013i	$-0.446699\pi$
$233$	5.08815	0.333336	0.166668	+	0.986013i	$0.446699\pi$
$234$	0	0
$235$	−4.26444	−0.278181
$236$	0	0
$237$	8.75600	0.568764
$238$	0	0
$239$	10.9239i	0.706611i	0.935508	+	0.353305i	$0.114942\pi$
$239$	10.9239i	0.706611i	−0.935508	+	0.353305i	$0.885058\pi$
$240$	0	0
$241$	− 11.9148i	− 0.767502i	−0.923437	−	0.383751i	$-0.874632\pi$
$241$	− 11.9148i	− 0.767502i	0.923437	−	0.383751i	$-0.125368\pi$
$242$	0	0
$243$	12.9855	0.833022
$244$	0	0
$245$	− 4.04892i	− 0.258676i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 4.28621i	− 0.271627i
$250$	0	0
$251$	22.3478	1.41058	0.705290	−	0.708919i	$-0.250817\pi$
$251$	22.3478	1.41058	0.705290	+	0.708919i	$0.250817\pi$
$252$	0	0
$253$	4.82908i	0.303602i
$254$	0	0
$255$	− 4.24698i	− 0.265956i
$256$	0	0
$257$	18.6601	1.16398	0.581992	−	0.813194i	$-0.302273\pi$
$257$	18.6601	1.16398	0.581992	+	0.813194i	$0.302273\pi$
$258$	0	0
$259$	10.9758	0.682005
$260$	0	0
$261$	−6.10752	−0.378046
$262$	0	0
$263$	−14.3991	−0.887887	−0.443944	−	0.896055i	$-0.646421\pi$
$263$	−14.3991	−0.887887	−0.443944	+	0.896055i	$0.646421\pi$
$264$	0	0
$265$	− 7.97823i	− 0.490099i
$266$	0	0
$267$	− 3.67025i	− 0.224616i
$268$	0	0
$269$	0.652793	0.0398015	0.0199007	−	0.999802i	$-0.493665\pi$
$269$	0.652793	0.0398015	0.0199007	+	0.999802i	$0.493665\pi$
$270$	0	0
$271$	− 1.99569i	− 0.121229i	−0.998161	−	0.0606147i	$-0.980694\pi$
$271$	− 1.99569i	− 0.121229i	0.998161	−	0.0606147i	$-0.0193061\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 7.43967i	− 0.448629i
$276$	0	0
$277$	−11.7845	−0.708061	−0.354030	−	0.935234i	$-0.615189\pi$
$277$	−11.7845	−0.708061	−0.354030	+	0.935234i	$0.615189\pi$
$278$	0	0
$279$	− 11.4916i	− 0.687982i
$280$	0	0
$281$	6.47219i	0.386098i	0.981189	+	0.193049i	$0.0618377\pi$
$281$	6.47219i	0.386098i	−0.981189	+	0.193049i	$0.938162\pi$
$282$	0	0
$283$	6.58104	0.391202	0.195601	−	0.980684i	$-0.437334\pi$
$283$	6.58104	0.391202	0.195601	+	0.980684i	$0.437334\pi$
$284$	0	0
$285$	4.69202	0.277931
$286$	0	0
$287$	−2.61058	−0.154098
$288$	0	0
$289$	11.0465	0.649796
$290$	0	0
$291$	6.61894i	0.388009i
$292$	0	0
$293$	24.3381i	1.42185i	0.703269	+	0.710924i	$0.251723\pi$
$293$	24.3381i	1.42185i	−0.703269	+	0.710924i	$0.748277\pi$
$294$	0	0
$295$	−17.6407	−1.02708
$296$	0	0
$297$	8.07069i	0.468309i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	12.5743i	0.724773i
$302$	0	0
$303$	−7.24996	−0.416500
$304$	0	0
$305$	− 12.3763i	− 0.708663i
$306$	0	0
$307$	− 14.0737i	− 0.803227i	−0.915809	−	0.401613i	$-0.868450\pi$
$307$	− 14.0737i	− 0.803227i	0.915809	−	0.401613i	$-0.131550\pi$
$308$	0	0
$309$	5.08815	0.289455
$310$	0	0
$311$	−29.7700	−1.68810	−0.844051	−	0.536263i	$-0.819836\pi$
$311$	−29.7700	−1.68810	−0.844051	+	0.536263i	$0.819836\pi$
$312$	0	0
$313$	−7.47889	−0.422732	−0.211366	−	0.977407i	$-0.567791\pi$
$313$	−7.47889	−0.422732	−0.211366	+	0.977407i	$0.567791\pi$
$314$	0	0
$315$	−7.97046	−0.449085
$316$	0	0
$317$	30.0301i	1.68666i	0.537396	+	0.843330i	$0.319408\pi$
$317$	30.0301i	1.68666i	−0.537396	+	0.843330i	$0.680592\pi$
$318$	0	0
$319$	− 5.79656i	− 0.324545i
$320$	0	0
$321$	3.82908	0.213719
$322$	0	0
$323$	30.9855i	1.72408i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.0677037i	0.00374402i
$328$	0	0
$329$	6.04652	0.333356
$330$	0	0
$331$	15.7168i	0.863872i	0.901904	+	0.431936i	$0.142169\pi$
$331$	15.7168i	0.863872i	−0.901904	+	0.431936i	$0.857831\pi$
$332$	0	0
$333$	14.4209i	0.790259i
$334$	0	0
$335$	−0.833397	−0.0455333
$336$	0	0
$337$	−1.95407	−0.106445	−0.0532224	−	0.998583i	$-0.516949\pi$
$337$	−1.95407	−0.106445	−0.0532224	+	0.998583i	$0.516949\pi$
$338$	0	0
$339$	4.05562	0.220271
$340$	0	0
$341$	10.9065	0.590619
$342$	0	0
$343$	20.0834i	1.08440i
$344$	0	0
$345$	1.51573i	0.0816041i
$346$	0	0
$347$	17.1250	0.919317	0.459659	−	0.888096i	$-0.347972\pi$
$347$	17.1250	0.919317	0.459659	+	0.888096i	$0.347972\pi$
$348$	0	0
$349$	10.4668i	0.560276i	0.959960	+	0.280138i	$0.0903802\pi$
$349$	10.4668i	0.560276i	−0.959960	+	0.280138i	$0.909620\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.5308i	0.826621i	0.910590	+	0.413310i	$0.135628\pi$
$353$	15.5308i	0.826621i	−0.910590	+	0.413310i	$0.864372\pi$
$354$	0	0
$355$	6.63879	0.352351
$356$	0	0
$357$	6.02177i	0.318706i
$358$	0	0
$359$	− 21.4263i	− 1.13083i	−0.824805	−	0.565417i	$-0.808715\pi$
$359$	− 21.4263i	− 1.13083i	0.824805	−	0.565417i	$-0.191285\pi$
$360$	0	0
$361$	−15.2325	−0.801711
$362$	0	0
$363$	2.48188	0.130265
$364$	0	0
$365$	15.2489	0.798164
$366$	0	0
$367$	−34.3032	−1.79061	−0.895306	−	0.445452i	$-0.853043\pi$
$367$	−34.3032	−1.79061	−0.895306	+	0.445452i	$0.853043\pi$
$368$	0	0
$369$	− 3.42998i	− 0.178557i
$370$	0	0
$371$	11.3123i	0.587305i
$372$	0	0
$373$	−12.5961	−0.652202	−0.326101	−	0.945335i	$-0.605735\pi$
$373$	−12.5961	−0.652202	−0.326101	+	0.945335i	$0.605735\pi$
$374$	0	0
$375$	− 6.34481i	− 0.327645i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 16.5386i	− 0.849529i	−0.905304	−	0.424765i	$-0.860357\pi$
$379$	− 16.5386i	− 0.849529i	0.905304	−	0.424765i	$-0.139643\pi$
$380$	0	0
$381$	−10.5278	−0.539356
$382$	0	0
$383$	− 7.53617i	− 0.385080i	−0.981289	−	0.192540i	$-0.938327\pi$
$383$	− 7.53617i	− 0.385080i	0.981289	−	0.192540i	$-0.0616726\pi$
$384$	0	0
$385$	− 7.56465i	− 0.385530i
$386$	0	0
$387$	−16.5211	−0.839815
$388$	0	0
$389$	−35.5555	−1.80274	−0.901369	−	0.433052i	$-0.857437\pi$
$389$	−35.5555	−1.80274	−0.901369	+	0.433052i	$0.857437\pi$
$390$	0	0
$391$	−10.0097	−0.506212
$392$	0	0
$393$	−1.80731	−0.0911670
$394$	0	0
$395$	− 22.7995i	− 1.14717i
$396$	0	0
$397$	− 1.35152i	− 0.0678308i	−0.999425	−	0.0339154i	$-0.989202\pi$
$397$	− 1.35152i	− 0.0678308i	0.999425	−	0.0339154i	$-0.0107977\pi$
$398$	0	0
$399$	−6.65279	−0.333056
$400$	0	0
$401$	− 0.579121i	− 0.0289199i	−0.999895	−	0.0144600i	$-0.995397\pi$
$401$	− 0.579121i	− 0.0289199i	0.999895	−	0.0144600i	$-0.00460291\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 9.13706i	− 0.454024i
$406$	0	0
$407$	−13.6866	−0.678422
$408$	0	0
$409$	− 15.1575i	− 0.749490i	−0.927128	−	0.374745i	$-0.877730\pi$
$409$	− 15.1575i	− 0.749490i	0.927128	−	0.374745i	$-0.122270\pi$
$410$	0	0
$411$	− 0.439665i	− 0.0216871i
$412$	0	0
$413$	25.0127	1.23079
$414$	0	0
$415$	−11.1608	−0.547860
$416$	0	0
$417$	6.29291	0.308165
$418$	0	0
$419$	35.7235	1.74521	0.872603	−	0.488430i	$-0.162430\pi$
$419$	35.7235	1.74521	0.872603	+	0.488430i	$0.162430\pi$
$420$	0	0
$421$	− 35.0465i	− 1.70806i	−0.520221	−	0.854032i	$-0.674151\pi$
$421$	− 35.0465i	− 1.70806i	0.520221	−	0.854032i	$-0.325849\pi$
$422$	0	0
$423$	7.94438i	0.386269i
$424$	0	0
$425$	15.4209	0.748022
$426$	0	0
$427$	17.5483i	0.849220i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	34.2814i	1.65128i	0.564199	+	0.825639i	$0.309185\pi$
$431$	34.2814i	1.65128i	−0.564199	+	0.825639i	$0.690815\pi$
$432$	0	0
$433$	−13.7385	−0.660232	−0.330116	−	0.943940i	$-0.607088\pi$
$433$	−13.7385	−0.660232	−0.330116	+	0.943940i	$0.607088\pi$
$434$	0	0
$435$	− 1.81940i	− 0.0872334i
$436$	0	0
$437$	− 11.0586i	− 0.529005i
$438$	0	0
$439$	10.2403	0.488742	0.244371	−	0.969682i	$-0.421419\pi$
$439$	10.2403	0.488742	0.244371	+	0.969682i	$0.421419\pi$
$440$	0	0
$441$	−7.54288	−0.359185
$442$	0	0
$443$	−12.1763	−0.578513	−0.289257	−	0.957252i	$-0.593408\pi$
$443$	−12.1763	−0.578513	−0.289257	+	0.957252i	$0.593408\pi$
$444$	0	0
$445$	−9.55688	−0.453039
$446$	0	0
$447$	4.66487i	0.220641i
$448$	0	0
$449$	12.9051i	0.609032i	0.952507	+	0.304516i	$0.0984947\pi$
$449$	12.9051i	0.609032i	−0.952507	+	0.304516i	$0.901505\pi$
$450$	0	0
$451$	3.25534	0.153288
$452$	0	0
$453$	7.84117i	0.368410i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.65710i	0.217850i	0.994050	+	0.108925i	$0.0347409\pi$
$457$	4.65710i	0.217850i	−0.994050	+	0.108925i	$0.965259\pi$
$458$	0	0
$459$	−16.7289	−0.780836
$460$	0	0
$461$	− 31.5405i	− 1.46899i	−0.678616	−	0.734493i	$-0.737420\pi$
$461$	− 31.5405i	− 1.46899i	0.678616	−	0.734493i	$-0.262580\pi$
$462$	0	0
$463$	17.6504i	0.820284i	0.912022	+	0.410142i	$0.134521\pi$
$463$	17.6504i	0.820284i	−0.912022	+	0.410142i	$0.865479\pi$
$464$	0	0
$465$	3.42327	0.158750
$466$	0	0
$467$	−32.1726	−1.48877	−0.744385	−	0.667751i	$-0.767257\pi$
$467$	−32.1726	−1.48877	−0.744385	+	0.667751i	$0.767257\pi$
$468$	0	0
$469$	1.18167	0.0545644
$470$	0	0
$471$	−5.23490	−0.241211
$472$	0	0
$473$	− 15.6799i	− 0.720964i
$474$	0	0
$475$	17.0368i	0.781704i
$476$	0	0
$477$	−14.8629	−0.680527
$478$	0	0
$479$	− 34.8998i	− 1.59461i	−0.603576	−	0.797306i	$-0.706258\pi$
$479$	− 34.8998i	− 1.59461i	0.603576	−	0.797306i	$-0.293742\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 2.14914i	− 0.0977895i
$484$	0	0
$485$	17.2349	0.782596
$486$	0	0
$487$	− 41.8351i	− 1.89573i	−0.318676	−	0.947864i	$-0.603238\pi$
$487$	− 41.8351i	− 1.89573i	0.318676	−	0.947864i	$-0.396762\pi$
$488$	0	0
$489$	4.83041i	0.218439i
$490$	0	0
$491$	21.8455	0.985873	0.492936	−	0.870065i	$-0.335924\pi$
$491$	21.8455	0.985873	0.492936	+	0.870065i	$0.335924\pi$
$492$	0	0
$493$	12.0151	0.541131
$494$	0	0
$495$	9.93900	0.446725
$496$	0	0
$497$	−9.41311	−0.422236
$498$	0	0
$499$	− 23.5472i	− 1.05412i	−0.849829	−	0.527058i	$-0.823295\pi$
$499$	− 23.5472i	− 1.05412i	0.849829	−	0.527058i	$-0.176705\pi$
$500$	0	0
$501$	− 13.2379i	− 0.591425i
$502$	0	0
$503$	7.08682	0.315986	0.157993	−	0.987440i	$-0.449498\pi$
$503$	7.08682	0.315986	0.157993	+	0.987440i	$0.449498\pi$
$504$	0	0
$505$	18.8780i	0.840060i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.61894i	0.337704i	0.985641	+	0.168852i	$0.0540059\pi$
$509$	7.61894i	0.337704i	−0.985641	+	0.168852i	$0.945994\pi$
$510$	0	0
$511$	−21.6213	−0.956471
$512$	0	0
$513$	− 18.4819i	− 0.815995i
$514$	0	0
$515$	− 13.2489i	− 0.583816i
$516$	0	0
$517$	−7.53989	−0.331604
$518$	0	0
$519$	10.4638	0.459311
$520$	0	0
$521$	−39.5133	−1.73111	−0.865555	−	0.500813i	$-0.833034\pi$
$521$	−39.5133	−1.73111	−0.865555	+	0.500813i	$0.833034\pi$
$522$	0	0
$523$	15.8194	0.691734	0.345867	−	0.938284i	$-0.387585\pi$
$523$	15.8194	0.691734	0.345867	+	0.938284i	$0.387585\pi$
$524$	0	0
$525$	3.31096i	0.144502i
$526$	0	0
$527$	22.6069i	0.984770i
$528$	0	0
$529$	−19.4276	−0.844678
$530$	0	0
$531$	32.8635i	1.42616i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 9.97046i	− 0.431061i
$536$	0	0
$537$	3.34183	0.144211
$538$	0	0
$539$	− 7.15883i	− 0.308353i
$540$	0	0
$541$	− 34.4819i	− 1.48249i	−0.671234	−	0.741246i	$-0.734235\pi$
$541$	− 34.4819i	− 1.48249i	0.671234	−	0.741246i	$-0.265765\pi$
$542$	0	0
$543$	2.65146	0.113785
$544$	0	0
$545$	0.176292	0.00755152
$546$	0	0
$547$	−36.8582	−1.57594	−0.787970	−	0.615713i	$-0.788868\pi$
$547$	−36.8582	−1.57594	−0.787970	+	0.615713i	$0.788868\pi$
$548$	0	0
$549$	−23.0562	−0.984015
$550$	0	0
$551$	13.2741i	0.565497i
$552$	0	0
$553$	32.3274i	1.37470i
$554$	0	0
$555$	−4.29590	−0.182351
$556$	0	0
$557$	− 1.27652i	− 0.0540879i	−0.999634	−	0.0270439i	$-0.991391\pi$
$557$	− 1.27652i	− 0.0540879i	0.999634	−	0.0270439i	$-0.00860940\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 7.50902i	− 0.317031i
$562$	0	0
$563$	−9.12737	−0.384673	−0.192336	−	0.981329i	$-0.561607\pi$
$563$	−9.12737	−0.384673	−0.192336	+	0.981329i	$0.561607\pi$
$564$	0	0
$565$	− 10.5603i	− 0.444277i
$566$	0	0
$567$	12.9554i	0.544075i
$568$	0	0
$569$	5.72156	0.239860	0.119930	−	0.992782i	$-0.461733\pi$
$569$	5.72156	0.239860	0.119930	+	0.992782i	$0.461733\pi$
$570$	0	0
$571$	7.60148	0.318112	0.159056	−	0.987270i	$-0.449155\pi$
$571$	7.60148	0.318112	0.159056	+	0.987270i	$0.449155\pi$
$572$	0	0
$573$	−10.2282	−0.427289
$574$	0	0
$575$	−5.50365	−0.229518
$576$	0	0
$577$	45.1564i	1.87989i	0.341330	+	0.939944i	$0.389123\pi$
$577$	45.1564i	1.87989i	−0.341330	+	0.939944i	$0.610877\pi$
$578$	0	0
$579$	− 3.35988i	− 0.139632i
$580$	0	0
$581$	15.8248	0.656522
$582$	0	0
$583$	− 14.1062i	− 0.584219i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.4040i	1.33746i	0.743507	+	0.668728i	$0.233161\pi$
$587$	32.4040i	1.33746i	−0.743507	+	0.668728i	$0.766839\pi$
$588$	0	0
$589$	−24.9758	−1.02911
$590$	0	0
$591$	− 6.33513i	− 0.260592i
$592$	0	0
$593$	− 36.6848i	− 1.50647i	−0.657754	−	0.753233i	$-0.728493\pi$
$593$	− 36.6848i	− 1.50647i	0.657754	−	0.753233i	$-0.271507\pi$
$594$	0	0
$595$	15.6799	0.642815
$596$	0	0
$597$	−7.71678	−0.315827
$598$	0	0
$599$	9.99223	0.408271	0.204136	−	0.978943i	$-0.434562\pi$
$599$	9.99223	0.408271	0.204136	+	0.978943i	$0.434562\pi$
$600$	0	0
$601$	−1.81163	−0.0738978	−0.0369489	−	0.999317i	$-0.511764\pi$
$601$	−1.81163	−0.0738978	−0.0369489	+	0.999317i	$0.511764\pi$
$602$	0	0
$603$	1.55257i	0.0632253i
$604$	0	0
$605$	− 6.46250i	− 0.262738i
$606$	0	0
$607$	−11.2161	−0.455248	−0.227624	−	0.973749i	$-0.573096\pi$
$607$	−11.2161	−0.455248	−0.227624	+	0.973749i	$0.573096\pi$
$608$	0	0
$609$	2.57971i	0.104535i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 20.8944i	− 0.843917i	−0.906615	−	0.421958i	$-0.861343\pi$
$613$	− 20.8944i	− 0.843917i	0.906615	−	0.421958i	$-0.138657\pi$
$614$	0	0
$615$	1.02177	0.0412018
$616$	0	0
$617$	12.0992i	0.487094i	0.969889	+	0.243547i	$0.0783110\pi$
$617$	12.0992i	0.487094i	−0.969889	+	0.243547i	$0.921689\pi$
$618$	0	0
$619$	− 10.5526i	− 0.424143i	−0.977254	−	0.212072i	$-0.931979\pi$
$619$	− 10.5526i	− 0.424143i	0.977254	−	0.212072i	$-0.0680210\pi$
$620$	0	0
$621$	5.97046	0.239586
$622$	0	0
$623$	13.5506	0.542895
$624$	0	0
$625$	−1.96184	−0.0784735
$626$	0	0
$627$	8.29590	0.331306
$628$	0	0
$629$	− 28.3696i	− 1.13117i
$630$	0	0
$631$	− 13.8514i	− 0.551417i	−0.961241	−	0.275709i	$-0.911087\pi$
$631$	− 13.8514i	− 0.551417i	0.961241	−	0.275709i	$-0.0889125\pi$
$632$	0	0
$633$	7.35019	0.292144
$634$	0	0
$635$	27.4131i	1.08786i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 12.3676i	− 0.489257i
$640$	0	0
$641$	−34.9608	−1.38087	−0.690434	−	0.723396i	$-0.742580\pi$
$641$	−34.9608	−1.38087	−0.690434	+	0.723396i	$0.742580\pi$
$642$	0	0
$643$	− 33.3980i	− 1.31709i	−0.752541	−	0.658545i	$-0.771172\pi$
$643$	− 33.3980i	− 1.31709i	0.752541	−	0.658545i	$-0.228828\pi$
$644$	0	0
$645$	− 4.92154i	− 0.193786i
$646$	0	0
$647$	2.32842	0.0915397	0.0457698	−	0.998952i	$-0.485426\pi$
$647$	2.32842	0.0915397	0.0457698	+	0.998952i	$0.485426\pi$
$648$	0	0
$649$	−31.1903	−1.22433
$650$	0	0
$651$	−4.85384	−0.190237
$652$	0	0
$653$	14.5714	0.570221	0.285111	−	0.958495i	$-0.407970\pi$
$653$	14.5714	0.570221	0.285111	+	0.958495i	$0.407970\pi$
$654$	0	0
$655$	4.70602i	0.183879i
$656$	0	0
$657$	− 28.4077i	− 1.10829i
$658$	0	0
$659$	−11.1395	−0.433932	−0.216966	−	0.976179i	$-0.569616\pi$
$659$	−11.1395	−0.433932	−0.216966	+	0.976179i	$0.569616\pi$
$660$	0	0
$661$	13.8498i	0.538694i	0.963043	+	0.269347i	$0.0868079\pi$
$661$	13.8498i	0.538694i	−0.963043	+	0.269347i	$0.913192\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	17.3230i	0.671759i
$666$	0	0
$667$	−4.28813	−0.166037
$668$	0	0
$669$	4.07069i	0.157382i
$670$	0	0
$671$	− 21.8823i	− 0.844757i
$672$	0	0
$673$	6.52973	0.251703	0.125851	−	0.992049i	$-0.459834\pi$
$673$	6.52973	0.251703	0.125851	+	0.992049i	$0.459834\pi$
$674$	0	0
$675$	−9.19806	−0.354034
$676$	0	0
$677$	−11.3104	−0.434693	−0.217346	−	0.976095i	$-0.569740\pi$
$677$	−11.3104	−0.434693	−0.217346	+	0.976095i	$0.569740\pi$
$678$	0	0
$679$	−24.4373	−0.937816
$680$	0	0
$681$	4.81402i	0.184474i
$682$	0	0
$683$	14.1793i	0.542555i	0.962501	+	0.271277i	$0.0874461\pi$
$683$	14.1793i	0.542555i	−0.962501	+	0.271277i	$0.912554\pi$
$684$	0	0
$685$	−1.14483	−0.0437418
$686$	0	0
$687$	7.59551i	0.289787i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 30.7952i	− 1.17151i	−0.810490	−	0.585753i	$-0.800799\pi$
$691$	− 30.7952i	− 1.17151i	0.810490	−	0.585753i	$-0.199201\pi$
$692$	0	0
$693$	−14.0925	−0.535328
$694$	0	0
$695$	− 16.3860i	− 0.621555i
$696$	0	0
$697$	6.74764i	0.255585i
$698$	0	0
$699$	2.82371	0.106802
$700$	0	0
$701$	−6.73184	−0.254258	−0.127129	−	0.991886i	$-0.540576\pi$
$701$	−6.73184	−0.254258	−0.127129	+	0.991886i	$0.540576\pi$
$702$	0	0
$703$	31.3424	1.18210
$704$	0	0
$705$	−2.36658	−0.0891307
$706$	0	0
$707$	− 26.7670i	− 1.00668i
$708$	0	0
$709$	47.6252i	1.78860i	0.447467	+	0.894300i	$0.352326\pi$
$709$	47.6252i	1.78860i	−0.447467	+	0.894300i	$0.647674\pi$
$710$	0	0
$711$	−42.4741	−1.59290
$712$	0	0
$713$	− 8.06829i	− 0.302160i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.06233i	0.226402i
$718$	0	0
$719$	−5.99330	−0.223512	−0.111756	−	0.993736i	$-0.535648\pi$
$719$	−5.99330	−0.223512	−0.111756	+	0.993736i	$0.535648\pi$
$720$	0	0
$721$	18.7855i	0.699610i
$722$	0	0
$723$	− 6.61224i	− 0.245912i
$724$	0	0
$725$	6.60627	0.245351
$726$	0	0
$727$	−24.1226	−0.894657	−0.447329	−	0.894370i	$-0.647625\pi$
$727$	−24.1226	−0.894657	−0.447329	+	0.894370i	$0.647625\pi$
$728$	0	0
$729$	−11.7627	−0.435656
$730$	0	0
$731$	32.5013	1.20210
$732$	0	0
$733$	36.0646i	1.33208i	0.745918	+	0.666038i	$0.232011\pi$
$733$	36.0646i	1.33208i	−0.745918	+	0.666038i	$0.767989\pi$
$734$	0	0
$735$	− 2.24698i	− 0.0828811i
$736$	0	0
$737$	−1.47352	−0.0542777
$738$	0	0
$739$	27.5254i	1.01254i	0.862375	+	0.506269i	$0.168976\pi$
$739$	27.5254i	1.01254i	−0.862375	+	0.506269i	$0.831024\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10.4692i	0.384078i	0.981387	+	0.192039i	$0.0615100\pi$
$743$	10.4692i	0.384078i	−0.981387	+	0.192039i	$0.938490\pi$
$744$	0	0
$745$	12.1468	0.445023
$746$	0	0
$747$	20.7918i	0.760731i
$748$	0	0
$749$	14.1371i	0.516557i
$750$	0	0
$751$	4.06770	0.148433	0.0742163	−	0.997242i	$-0.476354\pi$
$751$	4.06770	0.148433	0.0742163	+	0.997242i	$0.476354\pi$
$752$	0	0
$753$	12.4021	0.451957
$754$	0	0
$755$	20.4174	0.743066
$756$	0	0
$757$	20.4336	0.742670	0.371335	−	0.928499i	$-0.378900\pi$
$757$	20.4336	0.742670	0.371335	+	0.928499i	$0.378900\pi$
$758$	0	0
$759$	2.67994i	0.0972757i
$760$	0	0
$761$	27.0237i	0.979608i	0.871833	+	0.489804i	$0.162932\pi$
$761$	27.0237i	0.979608i	−0.871833	+	0.489804i	$0.837068\pi$
$762$	0	0
$763$	−0.249964	−0.00904929
$764$	0	0
$765$	20.6015i	0.744848i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 37.9407i	− 1.36818i	−0.729400	−	0.684088i	$-0.760201\pi$
$769$	− 37.9407i	− 1.36818i	0.729400	−	0.684088i	$-0.239799\pi$
$770$	0	0
$771$	10.3556	0.372947
$772$	0	0
$773$	− 16.3375i	− 0.587620i	−0.955864	−	0.293810i	$-0.905077\pi$
$773$	− 16.3375i	− 0.587620i	0.955864	−	0.293810i	$-0.0949232\pi$
$774$	0	0
$775$	12.4300i	0.446498i
$776$	0	0
$777$	6.09113	0.218518
$778$	0	0
$779$	−7.45473	−0.267093
$780$	0	0
$781$	11.7380	0.420017
$782$	0	0
$783$	−7.16660	−0.256114
$784$	0	0
$785$	13.6310i	0.486512i
$786$	0	0
$787$	− 18.6907i	− 0.666251i	−0.942882	−	0.333126i	$-0.891897\pi$
$787$	− 18.6907i	− 0.666251i	0.942882	−	0.333126i	$-0.108103\pi$
$788$	0	0
$789$	−7.99090	−0.284484
$790$	0	0
$791$	14.9734i	0.532394i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 4.42758i	− 0.157030i
$796$	0	0
$797$	−29.2519	−1.03615	−0.518077	−	0.855334i	$-0.673352\pi$
$797$	−29.2519	−1.03615	−0.518077	+	0.855334i	$0.673352\pi$
$798$	0	0
$799$	− 15.6286i	− 0.552901i
$800$	0	0
$801$	17.8039i	0.629068i
$802$	0	0
$803$	26.9614	0.951446
$804$	0	0
$805$	−5.59611	−0.197237
$806$	0	0
$807$	0.362273	0.0127526
$808$	0	0
$809$	−6.65087	−0.233832	−0.116916	−	0.993142i	$-0.537301\pi$
$809$	−6.65087	−0.233832	−0.116916	+	0.993142i	$0.537301\pi$
$810$	0	0
$811$	− 3.89200i	− 0.136667i	−0.997663	−	0.0683333i	$-0.978232\pi$
$811$	− 3.89200i	− 0.136667i	0.997663	−	0.0683333i	$-0.0217681\pi$
$812$	0	0
$813$	− 1.10752i	− 0.0388425i
$814$	0	0
$815$	12.5778	0.440581
$816$	0	0
$817$	35.9071i	1.25623i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	45.9982i	1.60535i	0.596418	+	0.802674i	$0.296590\pi$
$821$	45.9982i	1.60535i	−0.596418	+	0.802674i	$0.703410\pi$
$822$	0	0
$823$	7.95300	0.277224	0.138612	−	0.990347i	$-0.455736\pi$
$823$	7.95300	0.277224	0.138612	+	0.990347i	$0.455736\pi$
$824$	0	0
$825$	− 4.12870i	− 0.143743i
$826$	0	0
$827$	27.9648i	0.972432i	0.873839	+	0.486216i	$0.161623\pi$
$827$	27.9648i	0.972432i	−0.873839	+	0.486216i	$0.838377\pi$
$828$	0	0
$829$	−27.6310	−0.959665	−0.479833	−	0.877360i	$-0.659303\pi$
$829$	−27.6310	−0.959665	−0.479833	+	0.877360i	$0.659303\pi$
$830$	0	0
$831$	−6.53989	−0.226866
$832$	0	0
$833$	14.8388	0.514133
$834$	0	0
$835$	−34.4698	−1.19288
$836$	0	0
$837$	− 13.4843i	− 0.466085i
$838$	0	0
$839$	28.6848i	0.990311i	0.868804	+	0.495155i	$0.164889\pi$
$839$	28.6848i	0.990311i	−0.868804	+	0.495155i	$0.835111\pi$
$840$	0	0
$841$	−23.8528	−0.822509
$842$	0	0
$843$	3.59179i	0.123708i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9.16315i	0.314849i
$848$	0	0
$849$	3.65220	0.125343
$850$	0	0
$851$	10.1250i	0.347080i
$852$	0	0
$853$	43.2078i	1.47941i	0.672934	+	0.739703i	$0.265034\pi$
$853$	43.2078i	1.47941i	−0.672934	+	0.739703i	$0.734966\pi$
$854$	0	0
$855$	−22.7603	−0.778386
$856$	0	0
$857$	−35.1685	−1.20133	−0.600667	−	0.799499i	$-0.705098\pi$
$857$	−35.1685	−1.20133	−0.600667	+	0.799499i	$0.705098\pi$
$858$	0	0
$859$	−27.3793	−0.934168	−0.467084	−	0.884213i	$-0.654695\pi$
$859$	−27.3793	−0.934168	−0.467084	+	0.884213i	$0.654695\pi$
$860$	0	0
$861$	−1.44876	−0.0493737
$862$	0	0
$863$	41.3913i	1.40898i	0.709715	+	0.704489i	$0.248824\pi$
$863$	41.3913i	1.40898i	−0.709715	+	0.704489i	$0.751176\pi$
$864$	0	0
$865$	− 27.2465i	− 0.926409i
$866$	0	0
$867$	6.13036	0.208198
$868$	0	0
$869$	− 40.3116i	− 1.36748i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 32.1075i	− 1.08668i
$874$	0	0
$875$	23.4252	0.791916
$876$	0	0
$877$	− 24.7472i	− 0.835653i	−0.908527	−	0.417826i	$-0.862792\pi$
$877$	− 24.7472i	− 0.835653i	0.908527	−	0.417826i	$-0.137208\pi$
$878$	0	0
$879$	13.5066i	0.455567i
$880$	0	0
$881$	28.5875	0.963137	0.481568	−	0.876409i	$-0.340067\pi$
$881$	28.5875	0.963137	0.481568	+	0.876409i	$0.340067\pi$
$882$	0	0
$883$	9.61702	0.323639	0.161819	−	0.986820i	$-0.448264\pi$
$883$	9.61702	0.323639	0.161819	+	0.986820i	$0.448264\pi$
$884$	0	0
$885$	−9.78986	−0.329082
$886$	0	0
$887$	−15.9661	−0.536091	−0.268045	−	0.963406i	$-0.586378\pi$
$887$	−15.9661	−0.536091	−0.268045	+	0.963406i	$0.586378\pi$
$888$	0	0
$889$	− 38.8689i	− 1.30362i
$890$	0	0
$891$	− 16.1551i	− 0.541217i
$892$	0	0
$893$	17.2664	0.577797
$894$	0	0
$895$	− 8.70171i	− 0.290866i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.68473i	0.323004i
$900$	0	0
$901$	29.2392	0.974099
$902$	0	0
$903$	6.97823i	0.232221i
$904$	0	0
$905$	− 6.90408i	− 0.229500i
$906$	0	0
$907$	−28.8364	−0.957496	−0.478748	−	0.877952i	$-0.658909\pi$
$907$	−28.8364	−0.957496	−0.478748	+	0.877952i	$0.658909\pi$
$908$	0	0
$909$	35.1685	1.16647
$910$	0	0
$911$	−38.5633	−1.27766	−0.638830	−	0.769348i	$-0.720581\pi$
$911$	−38.5633	−1.27766	−0.638830	+	0.769348i	$0.720581\pi$
$912$	0	0
$913$	−19.7332	−0.653073
$914$	0	0
$915$	− 6.86831i	− 0.227059i
$916$	0	0
$917$	− 6.67264i	− 0.220350i
$918$	0	0
$919$	−8.87502	−0.292760	−0.146380	−	0.989228i	$-0.546762\pi$
$919$	−8.87502	−0.292760	−0.146380	+	0.989228i	$0.546762\pi$
$920$	0	0
$921$	− 7.81030i	− 0.257358i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 15.5985i	− 0.512875i
$926$	0	0
$927$	−24.6819	−0.810659
$928$	0	0
$929$	24.2295i	0.794945i	0.917614	+	0.397472i	$0.130113\pi$
$929$	24.2295i	0.794945i	−0.917614	+	0.397472i	$0.869887\pi$
$930$	0	0
$931$	16.3937i	0.537283i
$932$	0	0
$933$	−16.5211	−0.540877
$934$	0	0
$935$	−19.5526	−0.639437
$936$	0	0
$937$	17.2644	0.564005	0.282002	−	0.959414i	$-0.409001\pi$
$937$	17.2644	0.564005	0.282002	+	0.959414i	$0.409001\pi$
$938$	0	0
$939$	−4.15047	−0.135446
$940$	0	0
$941$	4.34050i	0.141496i	0.997494	+	0.0707482i	$0.0225387\pi$
$941$	4.34050i	0.141496i	−0.997494	+	0.0707482i	$0.977461\pi$
$942$	0	0
$943$	− 2.40821i	− 0.0784220i
$944$	0	0
$945$	−9.35258	−0.304240
$946$	0	0
$947$	− 45.0146i	− 1.46278i	−0.681961	−	0.731389i	$-0.738872\pi$
$947$	− 45.0146i	− 1.46278i	0.681961	−	0.731389i	$-0.261128\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	16.6655i	0.540415i
$952$	0	0
$953$	46.8859	1.51878	0.759391	−	0.650634i	$-0.225497\pi$
$953$	46.8859	1.51878	0.759391	+	0.650634i	$0.225497\pi$
$954$	0	0
$955$	26.6329i	0.861822i
$956$	0	0
$957$	− 3.21685i	− 0.103986i
$958$	0	0
$959$	1.62325	0.0524176
$960$	0	0
$961$	12.7778	0.412186
$962$	0	0
$963$	−18.5743	−0.598550
$964$	0	0
$965$	−8.74871	−0.281631
$966$	0	0
$967$	− 6.29457i	− 0.202420i	−0.994865	−	0.101210i	$-0.967729\pi$
$967$	− 6.29457i	− 0.202420i	0.994865	−	0.101210i	$-0.0322714\pi$
$968$	0	0
$969$	17.1957i	0.552404i
$970$	0	0
$971$	41.8068	1.34165	0.670823	−	0.741618i	$-0.265941\pi$
$971$	41.8068	1.34165	0.670823	+	0.741618i	$0.265941\pi$
$972$	0	0
$973$	23.2336i	0.744834i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	23.7530i	0.759926i	0.925002	+	0.379963i	$0.124063\pi$
$977$	23.7530i	0.759926i	−0.925002	+	0.379963i	$0.875937\pi$
$978$	0	0
$979$	−16.8974	−0.540043
$980$	0	0
$981$	− 0.328421i	− 0.0104857i
$982$	0	0
$983$	− 55.7251i	− 1.77736i	−0.458532	−	0.888678i	$-0.651625\pi$
$983$	− 55.7251i	− 1.77736i	0.458532	−	0.888678i	$-0.348375\pi$
$984$	0	0
$985$	−16.4959	−0.525602
$986$	0	0
$987$	3.35557	0.106809
$988$	0	0
$989$	−11.5996	−0.368845
$990$	0	0
$991$	35.5512	1.12932	0.564661	−	0.825323i	$-0.309007\pi$
$991$	35.5512	1.12932	0.564661	+	0.825323i	$0.309007\pi$
$992$	0	0
$993$	8.72215i	0.276789i
$994$	0	0
$995$	20.0935i	0.637007i
$996$	0	0
$997$	6.61058	0.209359	0.104680	−	0.994506i	$-0.466618\pi$
$997$	6.61058	0.209359	0.104680	+	0.994506i	$0.466618\pi$
$998$	0	0
$999$	16.9215i	0.535374i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2704.2.f.o.337.3		6
4.3	odd	2		169.2.b.b.168.1		6
12.11	even	2		1521.2.b.l.1351.6		6
13.5	odd	4		2704.2.a.z.1.2		3
13.8	odd	4		2704.2.a.ba.1.2		3
13.12	even	2	inner	2704.2.f.o.337.4		6
52.3	odd	6		169.2.e.b.147.6		12
52.7	even	12		169.2.c.b.146.1		6
52.11	even	12		169.2.c.b.22.1		6
52.15	even	12		169.2.c.c.22.3		6
52.19	even	12		169.2.c.c.146.3		6
52.23	odd	6		169.2.e.b.147.1		12
52.31	even	4		169.2.a.b.1.1	✓	3
52.35	odd	6		169.2.e.b.23.1		12
52.43	odd	6		169.2.e.b.23.6		12
52.47	even	4		169.2.a.c.1.3	yes	3
52.51	odd	2		169.2.b.b.168.6		6
156.47	odd	4		1521.2.a.o.1.1		3
156.83	odd	4		1521.2.a.r.1.3		3
156.155	even	2		1521.2.b.l.1351.1		6
260.99	even	4		4225.2.a.bb.1.1		3
260.239	even	4		4225.2.a.bg.1.3		3
364.83	odd	4		8281.2.a.bf.1.1		3
364.307	odd	4		8281.2.a.bj.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.a.b.1.1	✓	3	52.31	even	4
169.2.a.c.1.3	yes	3	52.47	even	4
169.2.b.b.168.1		6	4.3	odd	2
169.2.b.b.168.6		6	52.51	odd	2
169.2.c.b.22.1		6	52.11	even	12
169.2.c.b.146.1		6	52.7	even	12
169.2.c.c.22.3		6	52.15	even	12
169.2.c.c.146.3		6	52.19	even	12
169.2.e.b.23.1		12	52.35	odd	6
169.2.e.b.23.6		12	52.43	odd	6
169.2.e.b.147.1		12	52.23	odd	6
169.2.e.b.147.6		12	52.3	odd	6
1521.2.a.o.1.1		3	156.47	odd	4
1521.2.a.r.1.3		3	156.83	odd	4
1521.2.b.l.1351.1		6	156.155	even	2
1521.2.b.l.1351.6		6	12.11	even	2
2704.2.a.z.1.2		3	13.5	odd	4
2704.2.a.ba.1.2		3	13.8	odd	4
2704.2.f.o.337.3		6	1.1	even	1	trivial
2704.2.f.o.337.4		6	13.12	even	2	inner
4225.2.a.bb.1.1		3	260.99	even	4
4225.2.a.bg.1.3		3	260.239	even	4
8281.2.a.bf.1.1		3	364.83	odd	4
8281.2.a.bj.1.3		3	364.307	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2704.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.554958 q^{3} -1.44504i q^{5} +2.04892i q^{7} -2.69202 q^{9} -2.55496i q^{11} -0.801938i q^{15} +5.29590 q^{17} +5.85086i q^{19} +1.13706i q^{21} -1.89008 q^{23} +2.91185 q^{25} -3.15883 q^{27} +2.26875 q^{29} +4.26875i q^{31} -1.41789i q^{33} +2.96077 q^{35} -5.35690i q^{37} +1.27413i q^{41} +6.13706 q^{43} +3.89008i q^{45} -2.95108i q^{47} +2.80194 q^{49} +2.93900 q^{51} +5.52111 q^{53} -3.69202 q^{55} +3.24698i q^{57} -12.2078i q^{59} +8.56465 q^{61} -5.51573i q^{63} -0.576728i q^{67} -1.04892 q^{69} +4.59419i q^{71} +10.5526i q^{73} +1.61596 q^{75} +5.23490 q^{77} +15.7778 q^{79} +6.32304 q^{81} -7.72348i q^{83} -7.65279i q^{85} +1.25906 q^{87} -6.61356i q^{89} +2.36898i q^{93} +8.45473 q^{95} +11.9269i q^{97} +6.87800i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 6 q^{9} + 4 q^{17} - 10 q^{23} + 10 q^{25} - 2 q^{27} - 2 q^{29} - 8 q^{35} + 26 q^{43} + 8 q^{49} - 2 q^{51} + 2 q^{53} - 12 q^{55} + 8 q^{61} + 12 q^{69} + 30 q^{75} - 16 q^{77} + 10 q^{79}+ \cdots + 6 q^{95}+O(q^{100}) \) 6 * q + 4 * q^3 - 6 * q^9 + 4 * q^17 - 10 * q^23 + 10 * q^25 - 2 * q^27 - 2 * q^29 - 8 * q^35 + 26 * q^43 + 8 * q^49 - 2 * q^51 + 2 * q^53 - 12 * q^55 + 8 * q^61 + 12 * q^69 + 30 * q^75 - 16 * q^77 + 10 * q^79 - 2 * q^81 + 36 * q^87 + 6 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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